Examples of Cubical Homology in 2 Dimensions

On this page we take the topology examples previously introduced on the page here and go on to relate this to the homology of these examples.

Now we continue with the square example above but this time we treat it as a solid square. The mapping from edge to vertices is the same as above but now we also have a mapping from square to edges.

In order to represent the homology from this we need to have each mapping indexing into the next. cubical delta complex

So, on the above diagram going across the top row, starting from the square [1..2,1..2] we can generate all possible edges:

  • [1..1,1..2]
  • [2..2,1..2]
  • [1..2,1..1]
  • [1..2,2..2]

and from them we can generate all possible vertices:

  • [1..1,1..1]
  • [2..2,1..1]
  • [1..1,1..2]
  • [2..2,2..2]

Then, going left across the bottom row, we can replace these with indexes so that each dimension indexes into the next.

(1) -> ACUBE := FiniteCubicalComplex(Integer)

   (1)  FiniteCubicalComplex(Integer)
                                     Type: Type
(2) -> vs1:List(Integer) := [1,2,3,4]

   (2)  [1,2,3,4]
                            Type: List(Integer)
(3) -> sp := sphereSolid(2)$CubicalComplexFactory

   (3)
        (1..2,1..2)
            Type: FiniteCubicalComplex(Integer)
(4) -> d5 := sp::DeltaComplex(Integer)

   (4)
                  2D:[[- 1,4,2,- 3]]
         1D:[[- 1,2],[- 1,3],[- 2,4],[- 3,4]]
                 0D:[[0],[0],[0],[0]]
                    Type: DeltaComplex(Integer)

Conversion to Chain Complex

Now we have the delta complex we can go on to generate a 'chain complex', this consists of a sequence of matrices which map each array of indices to the next higher array.

I have put a more general discussion of chain complexes on page here.

Once we have added the implied faces we can then give each vertex an index number.

The index numbers, for each square, need not be consecutive as in the diagram but:

  • In each dimension the index numbers need to be asending.
  • For a given 'y' dimension all the indexes should be higher than all the indexes corresponding to a lower 'y' value and so on.
2d vertex

We can now map to a set of indexes for the edges as follows:

  • The edges are numbered in the same order as their source vertex.
  • If two edges have the same source vertex then the order is given by the target vertex.
  • The direction of the edge is in the same direction as the positive direction of the dimension it is moving in.
  • The source vertex is negative and the target is positive.
Edge Vertex
source target
1 -1 2
2 -1 3
3 -2 4
4 -3 4
This gives indexes for the edges as shown on the diagram on the right: 2d edge
 
Square Edge
1 -1 2 -3 4
  2d square

This matrix defines the mapping
from edges to squares.

Note: the edges are not numbered in the order that we travel around the square, that would be: 1,2,-4,-3.

   
square
 
edges  
[1..2,1..2]
 
1 =[1..2,1..1] left bracket
+1
right bracket
2 =[2..2,1..2]
+1
3 = -[1..1,1..2]
-1
4 = -[1..2,2..2]
-1

 

(5) -> chain(d5)

                     +- 1  - 1   0    0 + +- 1+
                     | 1    0   - 1   0 | | 1 |
   (5)  [0  0  0  0],|                  |,|   |,[]
                     | 0    1    0   - 1| |- 1|
                     + 0    0    1    1 + + 1 +
                             Type: ChainComplex
(6) -> homology(sp)

   (6)  [Z,0,0]
                           Type: List(Homology)

Another Example - Connected

If we add more squares like this:

two dimensional square
Then we get more columns in the matrix.   left bracket
squares
right bracket
edges
[1..2,1..2]
[2..3,1..2]
[3..4,1..2]
[1..2,1..1]
+1
0
0
[2..2,1..2]
+1
+1
+1
-[1..2,2..2]
-1
0
0
-[1..1,1..2]
-1
-1
-1
[2..3,1..1]
0
+1
0
[3..4,1..1]
0
0
+1
-[3..4,2..2]
0
0
-1
-[2..3,2..2]
0
-1
0

In FriCAS we can run as follows:

First we setup the cubical complex as described on previous page.

(7) -> Sq1 := cubicalFacet(1,[1..2,1..2])

   (7)  (1..2,1..2)
                             Type: CubicalFacet
(8) -> Sq2 := cubicalFacet(1,[2..3,1..2])

   (8)  (2..3,1..2)
                             Type: CubicalFacet
(9) -> Sq3 := cubicalFacet(1,[3..4,1..2])

   (9)  (3..4,1..2)
                             Type: CubicalFacet
(10) -> ex1:=cubicalComplex(vs1,[Sq1,Sq2,Sq3])$ACUBE

   (10)
         (1..2,1..2)
         (2..3,1..2)
         (3..4,1..2)
            Type: FiniteCubicalComplex(Integer)
(11) -> boundary(ex1)

   (11)
         -(1..1,1..2)
         (1..2,1..1)
         -(1..2,2..2)
         (2..3,1..1)
         -(2..3,2..2)
         (4..4,1..2)
         (3..4,1..1)
         -(3..4,2..2)
            Type: FiniteCubicalComplex(Integer

We can coerce this into a DeltaComplex which indexes each dimension into the next lower.

DeltaComplex is used internally for calculating chain and homology.

(12) -> d3 := ex1::DeltaComplex(Integer)

   (12)
   VCONCAT
      VCONCAT
          2D:[[- 1,4,2,- 3],[- 4,7,5,- 6],[- 7,10,8,- 9]]
     ,
          1D:
           [[- 1,2], [- 1,3], [- 2,4], [- 3,4], [- 3,5], [- 4,6], [- 5,6],
            [- 5,7], [- 6,8], [- 7,8]]
  ,
       0D:[[0],[0],[0],[0],[0],[0],[0],[0]]
                    Type: DeltaComplex(Integer)
The chain is matrix which goes from edges to to vertices.
(13) -> chain(d3)

   (13)
                              +- 1  - 1   0    0    0    0    0    0    0    0 +
                              | 1    0   - 1   0    0    0    0    0    0    0 |
                              | 0    1    0   - 1  - 1   0    0    0    0    0 |
                              | 0    0    1    1    0   - 1   0    0    0    0 |
     [0  0  0  0  0  0  0  0],|                                                |
                              | 0    0    0    0    1    0   - 1  - 1   0    0 |
                              | 0    0    0    0    0    1    1    0   - 1   0 |
                              | 0    0    0    0    0    0    0    1    0   - 1|
                              + 0    0    0    0    0    0    0    0    1    1 +
  ,
     +- 1   0    0 +
     | 1    0    0 |
     |- 1   0    0 |
     | 1   - 1   0 |
     |             | ++
     | 0    1    0 | ||
     |             |,||
     | 0   - 1   0 | ||
     |             | ++
     | 0    1   - 1|
     | 0    0    1 |
     | 0    0   - 1|
     + 0    0    1 +
                             Type: ChainComplex
Homology is connected space with no holes.
(14) -> homology(ex1)

   (14)  [Z,0,0]
                           Type: List(Homology)

Another Example - Not Connected

This time the squares don't connect to each other :

two dim seperated

In FriCAS we can run as follows:

First we setup the cubical complex as described on previous page.

(15) -> ex2:=cubicalComplex(vs1,[Sq1,Sq2,Sq3])$ACUBE

   (15)
         (1..2,1..2)
         (2..3,1..2)
         (3..4,1..2)
            Type: FiniteCubicalComplex(Integer)
(16) -> boundary(ex2)

   (16)
         -(1..1,1..2)
         (1..2,1..1)
         -(1..2,2..2)
         (2..3,1..1)
         -(2..3,2..2)
         (4..4,1..2)
         (3..4,1..1)
         -(3..4,2..2)
            Type: FiniteCubicalComplex(Integer)

We can coerce this into a DeltaComplex which indexes each dimension into the next lower.

DeltaComplex is used internally for calculating chain and homology.

(17) -> d32 := ex2::DeltaComplex(Integer)

   (17)
   VCONCAT
      VCONCAT
          2D:[[- 1,4,2,- 3],[- 4,7,5,- 6],[- 7,10,8,- 9]]
     ,
          1D:
           [[- 1,2], [- 1,3], [- 2,4], [- 3,4], [- 3,5], [- 4,6], [- 5,6],
            [- 5,7], [- 6,8], [- 7,8]]
  ,
       0D:[[0],[0],[0],[0],[0],[0],[0],[0]]
                    Type: DeltaComplex(Integer)
The chain is matrix which goes from edges to to vertices.
(18) -> chain(d32)

   (18)
                              +- 1  - 1   0    0    0    0    0    0    0    0 +
                              | 1    0   - 1   0    0    0    0    0    0    0 |
                              | 0    1    0   - 1  - 1   0    0    0    0    0 |
                              | 0    0    1    1    0   - 1   0    0    0    0 |
     [0  0  0  0  0  0  0  0],|                                                |
                              | 0    0    0    0    1    0   - 1  - 1   0    0 |
                              | 0    0    0    0    0    1    1    0   - 1   0 |
                               | 0    0    0    0    0    0    0    1    0   - 1|
                              + 0    0    0    0    0    0    0    0    1    1 +
  ,
     +- 1   0    0 +
     | 1    0    0 |
     |- 1   0    0 |
     | 1   - 1   0 |
     |             | ++
     | 0    1    0 | ||
     |             |,||
     | 0   - 1   0 | ||
     |             | ++
     | 0    1   - 1|
     | 0    0    1 |
     | 0    0   - 1|
     + 0    0    1 +
                             Type: ChainComplex
This time the space is not connected.
(19) -> homology(ex2)

   (19)  [Z,0,0]
                           Type: List(Homology)

Another Example - Squares with Hole

This time we leave a hole in the array of squares:

In FriCAS we can run as follows:

First we setup the cubical complex as described on previous page.

(20) -> Sq1 := cubicalFacet(1,[1..2,1..2])

   (20)  (1..2,1..2)
                             Type: CubicalFacet
(21) -> Sq2 := cubicalFacet(1,[2..3,1..2])

   (21)  (2..3,1..2)
                             Type: CubicalFacet
(22) -> Sq3 := cubicalFacet(1,[3..4,1..2])

   (22)  (3..4,1..2)
                             Type: CubicalFacet
(23) -> Sq4 := cubicalFacet(1,[1..2,2..3])

   (23)  (1..2,2..3)
                             Type: CubicalFacet
(24) -> Sq5 := cubicalFacet(1,[3..4,2..3])

   (24)  (3..4,2..3)
                             Type: CubicalFacet
(25) -> Sq6 := cubicalFacet(1,[1..2,3..4])

   (25)  (1..2,3..4)
                             Type: CubicalFacet
(26) -> Sq7 := cubicalFacet(1,[2..3,3..4])

   (26)  (2..3,3..4)
                             Type: CubicalFacet
(27) -> Sq8 := cubicalFacet(1,[3..4,3..4])

   (27)  (3..4,3..4)
                             Type: CubicalFacet
(28) -> sq1:=cubicalComplex(vs1,_
     [Sq1,Sq2,Sq3,Sq4,Sq5,Sq6,Sq7,Sq8])$ACUBE

   (28)
         (1..2,1..2)
         (2..3,1..2)
         (3..4,1..2)
         (1..2,2..3)
         (3..4,2..3)
         (1..2,3..4)
         (2..3,3..4)
         (3..4,3..4)
            Type: FiniteCubicalComplex(Integer)
(29) -> boundary(sq1)

   (29)
         -(1..1,1..2)
         (1..2,1..1)
         (2..3,1..1)
         -(2..3,2..2)
         (4..4,1..2)
         (3..4,1..1)
         -(1..1,2..3)
         (2..2,2..3)
         -(3..3,2..3)
         (4..4,2..3)
         -(1..1,3..4)
         -(1..2,4..4)
         (2..3,3..3)
         -(2..3,4..4)
         (4..4,3..4)
         -(3..4,4..4)
            Type: FiniteCubicalComplex(Integer)

We can coerce this into a DeltaComplex which indexes each dimension into the next lower.

DeltaComplex is used internally for calculating chain and homology.

(30) -> d3 := sq1::DeltaComplex(Integer)

   (30)
   VCONCAT
      VCONCAT
         VCONCAT
        ,
             2D:
              [[- 1,8,4,- 5], [- 2,9,5,- 6], [- 3,10,6,- 7], [- 8,15,11,- 12],
               [- 10,17,13,- 14], [- 15,22,18,- 19], [- 16,23,19,- 20],
               [- 17,24,20,- 21]]
     ,
          1D:
           [[- 1,2], [- 2,3], [- 3,4], [- 1,5], [- 2,6], [- 3,7], [- 4,8],
            [- 5,6], [- 6,7], [- 7,8], [- 5,9], [- 6,10], [- 7,11], [- 8,12],
            [- 9,10], [- 10,11], [- 11,12], [- 9,13], [- 10,14], [- 11,15],
            [- 12,16], [- 13,14], [- 14,15], [- 15,16]]
  ,
       0D:[[0],[0],[0],[0],[0],[0],[0],[0],[0],[0],[0],[0],[0],[0],[0],[0]]
                    Type: DeltaComplex(Integer)
The chain is matrix which goes from edges to to vertices.
(31) -> chain(d3)

   (31)
     [0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
  ,
   [[- 1,0,0,- 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
    [1,- 1,0,0,- 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
    [0,1,- 1,0,0,- 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
    [0,0,1,0,0,0,- 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
    [0,0,0,1,0,0,0,- 1,0,0,- 1,0,0,0,0,0,0,0,0,0,0,0,0,0],
    [0,0,0,0,1,0,0,1,- 1,0,0,- 1,0,0,0,0,0,0,0,0,0,0,0,0],
    [0,0,0,0,0,1,0,0,1,- 1,0,0,- 1,0,0,0,0,0,0,0,0,0,0,0],
    [0,0,0,0,0,0,1,0,0,1,0,0,0,- 1,0,0,0,0,0,0,0,0,0,0],
    [0,0,0,0,0,0,0,0,0,0,1,0,0,0,- 1,0,0,- 1,0,0,0,0,0,0],
    [0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,- 1,0,0,- 1,0,0,0,0,0],
    [0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,- 1,0,0,- 1,0,0,0,0],
    [0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,- 1,0,0,0],
    [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,- 1,0,0],
    [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,- 1,0],
    [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,- 1],
    [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1]]
  ,
     +- 1   0    0    0    0    0    0    0 +
     | 0   - 1   0    0    0    0    0    0 |
     | 0    0   - 1   0    0    0    0    0 |
     | 1    0    0    0    0    0    0    0 |
     |- 1   1    0    0    0    0    0    0 |
     | 0   - 1   1    0    0    0    0    0 |
     | 0    0   - 1   0    0    0    0    0 |
     | 1    0    0   - 1   0    0    0    0 |
     | 0    1    0    0    0    0    0    0 | ++
     | 0    0    1    0   - 1   0    0    0 | ||
     | 0    0    0    1    0    0    0    0 | ||
     | 0    0    0   - 1   0    0    0    0 | ||
     |                                      |,||
     | 0    0    0    0    1    0    0    0 | ||
     | 0    0    0    0   - 1   0    0    0 | ||
     | 0    0    0    1    0   - 1   0    0 | ||
     | 0    0    0    0    0    0   - 1   0 | ++
     | 0    0    0    0    1    0    0   - 1|
     | 0    0    0    0    0    1    0    0 |
     | 0    0    0    0    0   - 1   1    0 |
     | 0    0    0    0    0    0   - 1   1 |
     | 0    0    0    0    0    0    0   - 1|
     | 0    0    0    0    0    1    0    0 |
     | 0    0    0    0    0    0    1    0 |
     + 0    0    0    0    0    0    0    1 +
                             Type: ChainComplex
This time the space is not connected.
(32) -> homology(sq1)

   (32)  [Z,Z,0]
                           Type: List(Homology)

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  • I have put the code here.
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